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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m}, k)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k = R/\\mathfrak{m}$.\nLet $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module.\nFor each $i \\ge 0$, the module $\\operatorname{Tor}^R_i(M, k)$ is annihilated by $\\mathfrak{m}$, so it is a finite-dimensional $k$-vector space.\nThe $i$-th *Betti number* of $M$ is\n$$\\beta_i(M) = \\dim_k \\operatorname{Tor}^R_i(M, k).$$\n\nIn the graded case, $R = k[x_1, \\ldots, x_n]$ is graded by degree, $\\mathfrak{m} = (x_1, \\ldots, x_n)$, and $M$ is a finitely generated graded $R$-module.\nThen each $\\operatorname{Tor}^R_i(M, k)$ is graded, and the *graded Betti numbers* are $\\beta_{i,j}(M) = \\dim_k \\operatorname{Tor}^R_i(M, k)_j$.\n\n## Remarks\n\nEquivalently, $\\beta_i(M)$ is the rank of the [[free module|0DZ9DKAKZR]] $F_i$ in a minimal free [[resolution|HK2AA1YSQ2]] $\\cdots \\to F_1 \\to F_0 \\to M \\to 0$, that is, one whose differentials have all entries in $\\mathfrak{m}$.\n\nFor $R = k[x, y]$ and $M = k = R/(x, y)$, the Koszul complex\n$$0 \\to R(-2) \\to R(-1)^2 \\to R \\to k \\to 0$$\nis a minimal resolution, so $\\beta_0 = 1$, $\\beta_1 = 2$, $\\beta_2 = 1$, and the nonzero graded Betti numbers are $\\beta_{0,0} = 1$, $\\beta_{1,1} = 2$, $\\beta_{2,2} = 1$.\n\nThe ranks in a resolution that is not minimal are not the Betti numbers.\nFor $M = R/(x)$ over $R = k[x]$, the resolution $0 \\to R \\oplus R \\to R \\oplus R \\to M \\to 0$, with the second summand mapped identically, has ranks $2, 2$, while $\\beta_0 = \\beta_1 = 1$.\n\n$\\beta_0(M)$ is the least number of generators of $M$, and the largest $i$ with $\\beta_i(M) \\ne 0$ is the [[projective dimension|MR9QCH1PNA]] of $M$ when it is finite.\n",
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